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Suppose that A, B, and AB are normal matrices. Prove that BA is also normal.

Here some hints: the trace of matrices can be used in clever ways to prove equalities.
Note that tr(A+B)=tr(A) + tr(B), tr(AB)=tr(BA) for any square A and B, and tr (C*C) 0 with equality if and only if C=0.There may be other techniques for doing this as well .
You can use to the theorem : Let A a square matrix then A is normal if only if there is a polynomial g such that A*=g(A). Where A* is the conjugate transpose.

I don't know what is better for you but when you prove this problem please be sure to use
the appropriate theorem or tool and explain with detail the prove, Thank you

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